Also, by unchecking Vsync, what is it really doing?
- - rendering frames faster than the LCD can refresh?
- - Or is it trying to force the LCD to refresh at a higher frequency, which I assume could damage it?

First of all, don't use Vsync, use the frame limit option in config->cpu.
Vsync put a lot of stress on the GPU and in some cases will reduce your fps and/or introduce input lag.
Vsync does not limit fps to 60. Instead, it synchronizes the frames to the refresh cycles of your monitor.

So let's say your fps is actually higher than 60. Vsync will tell your GPU to stop at 60 frames, store the remaining frames in VRAM (video RAM), and wait for the next refresh cycle of the monitor. The problem is you only have a limited amount of VRAM and an increasing amount of frames to be stored. As a result, you get less fps, because your GPU have little to no VRAM to work with.

The frame limit option in cpu is different. It literally tells the emulator to render at a specific amount of fps. The default is 60, but you can change it to whatever.

Onwards to the other question. No, running at a frame rate higher than your monitor refresh rate won't damage your monitor. What happens is your GPU gives a whole bunch of frames for your monitor to display. If the monitor does not refresh in time for all of the frames, it will display whatever it refreshes in time for (i.e. half of this frame and half of that frame, or whatever combination you can think of). That is called tearing. Vsync removes tearing, but is demanding.

And that covers that. Phew.

Almost forgot. It's 60Hz not MHz, because then that would be 60,000,000 fps.

Nappa: Vegeta! What does the scouter say about his power level?
Vegeta: It's...one thousand and six.
Nappa: Wh-...really?
Vegeta: Yeah! Beat him up Nappa!
Nappa: Yay!

The Good Doctor pretty much summed it up.
All I have to add are some quirks with V-Sync that are specific to PCSX2.

I have 1 GB of RAM on my Vee Dee Ooh (That's Video) card.
I run GSdx @ 3x. (The same thing described below happens @ native, too)

In just about every RPG made for the PS2 and released in America, enabling V-Sync causes major slowdowns when the polygon count gets too high, on my PC.
All I own are RPGs...

Anyway, in some games, this can be counteracted by setting the SpeedHack called VU Cycle Stealing to "2". However, it's not 100% effective and can cause other issues, like the game crashing, or producing garbage onscreen. Also, sometimes you'll still get FPS drops.

Also, for some reason, FFX in particular HATES having v-sync on.
It messes with the spoken dialogue, and causes major slowdowns in many areas.

However, don't lose hope!
I'm confident that a future revision of PCSX2, and/or a new Graphics Plugin will be made, that will eliminate screen tearing in games that didn't exhibit such behavior when played on the actual PS2 Hardware.
It's only a matter of time.

Actually vsync is a signal that forces the video card to match the monitor sweeping value. That's to grant the image will be correct without (I can't recall the correct word) partial sliding and shifts. In a way it's more a memento of the CRT tubes days and used on TV standards.

It would not prevent the card to write it's buffer more times if it can be done without losing the sync. The bad news is when the video card can't achieve (fill the buffer) the nominal value (60hz in that case) so it will refrain from writing it's buffer until it can be done together with the vsync signal, that means it will reduce the FPS by a greater amount, not seldom to half that nominal value (let's say it will write at each 2 vsync signals, 30FPS in this case).

If the game (any graphic application actually) presents image distortions it may be need to use the vsync but it's not that common and normally it should be kept set off.

The bottom line it's the Vsync can't force the FPS up but almost certainly will force it down. The only reason for it is to grant the image correctness, what is not always a real necessity.

I noticed myself in Split Second Velocity if I turn on VSync the screen got way more saturation, while turned off it's gone, but V-Sync has to do indeed that it locks your fps on PC Games towards 60, 75 or whatever the refresh rate is of your monitor.